Pub Date : 2014-01-01DOI: 10.1112/S1461157013000351
I. Krasikov
We show how one can obtain an asymptotic expression for some special functions with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function Jν(x) and the Airy function Ai(x). In particular, we answer a question raised by Olenko and find a sharp bound on the difference between Jν(x) and its standard asymptotics. We also give a very simple and surprisingly precise approximation for the zeros Ai(x).
{"title":"Approximations for the Bessel and Airy functions with an explicit error term","authors":"I. Krasikov","doi":"10.1112/S1461157013000351","DOIUrl":"https://doi.org/10.1112/S1461157013000351","url":null,"abstract":"We show how one can obtain an asymptotic expression for some special functions with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function Jν(x) and the Airy function Ai(x). In particular, we answer a question raised by Olenko and find a sharp bound on the difference between Jν(x) and its standard asymptotics. We also give a very simple and surprisingly precise approximation for the zeros Ai(x).","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"209-225"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000351","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000230
S. Blackburn, Samuel Scott
Let G be a cyclic group written multiplicatively (and represented in some concrete way). Let n be a positive integer (much smaller than the order of G). Let g, h ∈ G. The bounded height discrete logarithm problem is the task of finding positive integers a and b (if they exist) such that a 6 n, b 6 n and g = h. (Provided that b is coprime to the order of g, we have h = g where a/b is a rational number of height at most n. This motivates the terminology.) The paper provides a reduction to the two-dimensional discrete logarithm problem, so the bounded height discrete logarithm problem can be solved using a low-memory heuristic algorithm for the two-dimensional discrete logarithm problem due to Gaudry and Schost. The paper also provides a low-memory heuristic algorithm to solve the bounded height discrete logarithm problem in a generic group directly, without using a reduction to the two-dimensional discrete logarithm problem. This new algorithm is inspired by (but differs from) the Gaudry– Schost algorithm. Both algorithms use O(n) group operations, but the new algorithm is faster and simpler than the Gaudry–Schost algorithm when used to solve the bounded height discrete logarithm problem. Like the Gaudry–Schost algorithm, the new algorithm can easily be carried out in a distributed fashion. The bounded height discrete logarithm problem is relevant to a class of attacks on the privacy of a key establishment protocol recently published by EMVCo for comment. This protocol is intended to protect the communications between a chip-based payment card and a terminal using elliptic curve cryptography. The paper comments on the implications of these attacks for the design of any final version of the EMV protocol.
{"title":"The discrete logarithm problem for exponents of bounded height","authors":"S. Blackburn, Samuel Scott","doi":"10.1112/S1461157014000230","DOIUrl":"https://doi.org/10.1112/S1461157014000230","url":null,"abstract":"Let G be a cyclic group written multiplicatively (and represented in some concrete way). Let n be a positive integer (much smaller than the order of G). Let g, h ∈ G. The bounded height discrete logarithm problem is the task of finding positive integers a and b (if they exist) such that a 6 n, b 6 n and g = h. (Provided that b is coprime to the order of g, we have h = g where a/b is a rational number of height at most n. This motivates the terminology.) The paper provides a reduction to the two-dimensional discrete logarithm problem, so the bounded height discrete logarithm problem can be solved using a low-memory heuristic algorithm for the two-dimensional discrete logarithm problem due to Gaudry and Schost. The paper also provides a low-memory heuristic algorithm to solve the bounded height discrete logarithm problem in a generic group directly, without using a reduction to the two-dimensional discrete logarithm problem. This new algorithm is inspired by (but differs from) the Gaudry– Schost algorithm. Both algorithms use O(n) group operations, but the new algorithm is faster and simpler than the Gaudry–Schost algorithm when used to solve the bounded height discrete logarithm problem. Like the Gaudry–Schost algorithm, the new algorithm can easily be carried out in a distributed fashion. The bounded height discrete logarithm problem is relevant to a class of attacks on the privacy of a key establishment protocol recently published by EMVCo for comment. This protocol is intended to protect the communications between a chip-based payment card and a terminal using elliptic curve cryptography. The paper comments on the implications of these attacks for the design of any final version of the EMV protocol.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"148-156"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000230","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-11-27DOI: 10.1112/S1461157015000182
M. Bennett, Amir Ghadermarzi
We solve the Diophantine equation $Y^{2}=X^{3}+k$ for all nonzero integers $k$ with $|k|leqslant 10^{7}$ . Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.
{"title":"Mordell’s equation: a classical approach","authors":"M. Bennett, Amir Ghadermarzi","doi":"10.1112/S1461157015000182","DOIUrl":"https://doi.org/10.1112/S1461157015000182","url":null,"abstract":"We solve the Diophantine equation $Y^{2}=X^{3}+k$\u0000 for all nonzero integers $k$\u0000 with $|k|leqslant 10^{7}$\u0000 . Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"633-646"},"PeriodicalIF":0.0,"publicationDate":"2013-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000182","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-11-24DOI: 10.1112/S1461157015000145
M. Riza, Hatice Aktore
This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge–Kutta method, derived in the framework of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus on positive-valued functions of real variables and the fact that the multiplicative derivative does not exist at the roots of the function are presented explicitly to ensure that the proposed method is universally applicable. The error and stability analyses are also carried out explicitly in the framework of geometric multiplicative calculus. The method presented is applied to various problems and the results are compared to those obtained from the ordinary Runge–Kutta method. Moreover, for one example, a comparison of the computation time against relative error is worked out to illustrate the general advantage of the proposed method.
{"title":"The Runge-Kutta method in geometric multiplicative calculus","authors":"M. Riza, Hatice Aktore","doi":"10.1112/S1461157015000145","DOIUrl":"https://doi.org/10.1112/S1461157015000145","url":null,"abstract":"This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge–Kutta method, derived in the framework of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus on positive-valued functions of real variables and the fact that the multiplicative derivative does not exist at the roots of the function are presented explicitly to ensure that the proposed method is universally applicable. The error and stability analyses are also carried out explicitly in the framework of geometric multiplicative calculus. The method presented is applied to various problems and the results are compared to those obtained from the ordinary Runge–Kutta method. Moreover, for one example, a comparison of the computation time against relative error is worked out to illustrate the general advantage of the proposed method.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"539-554"},"PeriodicalIF":0.0,"publicationDate":"2013-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000145","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-11-08DOI: 10.1112/S1461157014000084
M. Klug, Michael D. Musty, S. Schiavone, J. Voight
We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.
{"title":"Numerical calculation of three-point branched covers of the projective line","authors":"M. Klug, Michael D. Musty, S. Schiavone, J. Voight","doi":"10.1112/S1461157014000084","DOIUrl":"https://doi.org/10.1112/S1461157014000084","url":null,"abstract":"We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"379-430"},"PeriodicalIF":0.0,"publicationDate":"2013-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000084","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-10-01DOI: 10.1112/S1461157013000211
V. Dabbaghian, J. Dixon
The so-called Burnside-Dixon-Schneider (BDS) method currently used as the default method of computing character tables in GAP for groups which are not solvable is often inecient in dealing with groups with large centres. If G is a nite group with centre Z and a linear character of Z, then we describe a method of computing the set Irr(G; ) of irreducible characters of G whose restriction Z is a multiple of . This modication of the BDS method involves only jIrr(G; )j conjugacy classes of G and so is relatively fast. A generalization of the method can be applied to computation of small sets of characters of groups with a solvable normal subgroup.
{"title":"Computing characters of groups with central subgroups","authors":"V. Dabbaghian, J. Dixon","doi":"10.1112/S1461157013000211","DOIUrl":"https://doi.org/10.1112/S1461157013000211","url":null,"abstract":"The so-called Burnside-Dixon-Schneider (BDS) method currently used as the default method of computing character tables in GAP for groups which are not solvable is often inecient in dealing with groups with large centres. If G is a nite group with centre Z and a linear character of Z, then we describe a method of computing the set Irr(G; ) of irreducible characters of G whose restriction Z is a multiple of . This modication of the BDS method involves only jIrr(G; )j conjugacy classes of G and so is relatively fast. A generalization of the method can be applied to computation of small sets of characters of groups with a solvable normal subgroup.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"398-406"},"PeriodicalIF":0.0,"publicationDate":"2013-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000211","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-07-23DOI: 10.1112/S1461157014000072
P. L. Clark, Patrick Corn, Alex Rice, James Stankewicz
We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number elds of degree 1-13. Addi- tionally we describe the algorithm used to compute these torsion subgroups and its implementation.
{"title":"COMPUTATION ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION","authors":"P. L. Clark, Patrick Corn, Alex Rice, James Stankewicz","doi":"10.1112/S1461157014000072","DOIUrl":"https://doi.org/10.1112/S1461157014000072","url":null,"abstract":"We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number elds of degree 1-13. Addi- tionally we describe the algorithm used to compute these torsion subgroups and its implementation.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"509-535"},"PeriodicalIF":0.0,"publicationDate":"2013-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000072","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-07-02DOI: 10.1112/S1461157013000338
Thomas Hamilton, David Loeffler
We give a computationally effective criterion for determining whether a finite-index subgroup of SL2(Z) is a congruence subgroup, extending earlier work of Hsu for subgroups of PSL2(Z).
{"title":"Congruence testing for odd subgroups of the modular group","authors":"Thomas Hamilton, David Loeffler","doi":"10.1112/S1461157013000338","DOIUrl":"https://doi.org/10.1112/S1461157013000338","url":null,"abstract":"We give a computationally effective criterion for determining whether a finite-index subgroup of SL2(Z) is a congruence subgroup, extending earlier work of Hsu for subgroups of PSL2(Z).","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"206-208"},"PeriodicalIF":0.0,"publicationDate":"2013-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000338","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-07-01DOI: 10.1112/S1461157015000121
F. Bouyer, Marco Streng
Van Wamelen [Math. Comp. 68 (1999) no. 225, 307–320] lists 19 curves of genus two over $mathbf{Q}$ with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over $mathbf{Q}$ , and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field. We extend Van Wamelen’s list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest ‘generic’ examples of CM curves of genus two. We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.
Van Wamelen[数学]第68(1999)号22,307 - 320]用复乘法(CM)列出了$mathbf{Q}$上的19条2属曲线。然而，对于每条曲线，CM-field都是$mathbf{Q}$上的循环伽罗瓦，而非伽罗瓦四次CM-field的一般情况并没有出现在这个列表中。原因是在这种情况下定义的场总是包含反射场的实二次子场。我们将Van Wamelen列表扩展到包含在这个实数二次域上定义的2属曲线。因此，我们的列表包含了最小的2属CM曲线的“一般”例子。我们解释了获得这个列表的方法，包括一种新的全实数域上任意超椭圆曲线的高度降低算法。与Van Wamelen不同的是，我们还通过实现Lauter和Viray对Igusa类多项式的分母界来证明我们的列表。
{"title":"Examples of CM curves of genus two defined over the reflex field","authors":"F. Bouyer, Marco Streng","doi":"10.1112/S1461157015000121","DOIUrl":"https://doi.org/10.1112/S1461157015000121","url":null,"abstract":"Van Wamelen [Math. Comp. 68 (1999) no. 225, 307–320] lists 19 curves of genus two over $mathbf{Q}$\u0000 with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over \u0000 $mathbf{Q}$\u0000 , and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field.\u0000 We extend Van Wamelen’s list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest ‘generic’ examples of CM curves of genus two.\u0000 We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"507-538"},"PeriodicalIF":0.0,"publicationDate":"2013-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000121","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-06-01DOI: 10.1112/S1461157013000077
John W. Jones
For each solvable Galois group which appears in degree 9 and each allowable signature, we nd polynomials which dene the elds of minimum absolute discriminant.
对于每一个出现在9次的可解伽罗瓦群和每一个允许签名，我们得到了满足最小绝对判别式域的多项式。
{"title":"Minimal solvable nonic fields","authors":"John W. Jones","doi":"10.1112/S1461157013000077","DOIUrl":"https://doi.org/10.1112/S1461157013000077","url":null,"abstract":"For each solvable Galois group which appears in degree 9 and each allowable signature, we nd polynomials which dene the elds of minimum absolute discriminant.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"130-138"},"PeriodicalIF":0.0,"publicationDate":"2013-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000077","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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